Theorem tz6.26i

Description: All nonempty subclasses of a class having a well-ordered set-like relation R have R-minimal elements. Proposition 6.26 of TakeutiZaring p. 31. (Contributed by Scott Fenton, 14-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

	Ref	Expression
Hypotheses	tz6.26i.1	$$⊢R\mathrm{We}A$$
	tz6.26i.2	$$⊢R\mathrm{Se}A$$
Assertion	tz6.26i	$$⊢\left(B\subseteq A\wedge B\ne \varnothing \right)\to \exists y\in BPred\left(R,B,y\right)=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		tz6.26i.1	$$⊢R\mathrm{We}A$$
2		tz6.26i.2	$$⊢R\mathrm{Se}A$$
3		tz6.26	$$⊢\left(\left(R\mathrm{We}A\wedge R\mathrm{Se}A\right)\wedge \left(B\subseteq A\wedge B\ne \varnothing \right)\right)\to \exists y\in BPred\left(R,B,y\right)=\varnothing$$
4	1 2 3	mpanl12	$$⊢\left(B\subseteq A\wedge B\ne \varnothing \right)\to \exists y\in BPred\left(R,B,y\right)=\varnothing$$